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Angles in Standard Position

Sine Law & Ambiguous Case

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UNIT 6 : BASIC TRIGONOMETRY WITH TRIANGLES

LESSON 3: ANGLES IN STANDARD POSITION

Definition: An angle is in standard position if it has its vertex at the origin and initial arm along the positive x-axis. The terminal arm is found by rotating the initial arm about the origin to a terminal position in one of the 4 quadrants. The rotation is positive if it is in the counter – clockwise direction and negative if in the clockwise direction.

Example 1:

Solution:

x = 4

y = 3

r = 5

Example 2: Angles greater than 90⁰

Solution:

x = -5

y = 12

r = 13

Note that sine is positive and cosine and tangent are negative for a second quadrant angle.

Example 3: Complementary angles (angles which add to 180^o).

Using your calculator, sin 120^o = 0.8660254; also sin 60^o = 0.8660254

Hence sin 120^o = sin 60^o or

sin(120^o) = sin (180^o – 120^o) = sin 60^o

Note that sine is positive in both the first and second quadrants.

Note 60^o is the reference(related acute) angle relative to 120^o.

Definition:

The reference (related acute) angle is the angle between the terminal arm and the x-axis.

In the above example (120^o), it is found by subtracting from 180^o.

Similarly, cos 150^o = -0.8660254; also cos 30 = 0.8660254

Hence cos 150^o = -cos 30⁰ or

cos(150^o) = -cos (180^o – 150^o) = -cos 30^o

Note that cosine is positive in the first quadrant and negative in the second quadrant.

Note 30^o is the reference (related acute) angle relative to 150^o (180^o – 150^o = 30^o)

Result: To find the trigonometric ratios of angles between 90^o and 180^o, use the following rules:

Examples:

1. Find cos 135^o

Solution:

cos 135^o = - cos (180^o – 135^o)

= - cos 45^o ** 45^o is reference (related acute) angle

= - 0.707106781 ** Using your cacluator

2. Find sin 114^o.

Solution:

sin 114^o = sin(180^o – 114^o)

= sin 66^o

= 0.913545457

3. Find tan 161^o

Solution:

tan 161^o = - tan(180^o – 161^o)

= - tan 19^o

= - 0.344327613

Note that tangent is positive in the first quadrant and negative in the second quadrant.

4. Find sin 123^o using a calculator to 5 decimal places.

Solution:

5. Find sec 149^o using the calculator to 5 decimal places.

Solution:

Solution:

Solution:

Since csc A is positive, < A could be in either the first or second quadrant. There are two solutions.

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